Brolin’s equidistribution theorem in \(p\)-adic dynamics. (Théorème d’équidistribution de Brolin en dynamique \(p\)-adique.) (French. Abridged English version) Zbl 1052.37039

Favre, Charles; Rivera-Letelier, Juan

C. R., Math., Acad. Sci. Paris 339, No. 4, 271-276 (2004).

Summary: We prove an analog of the famous equidistribution theorem of Brolin for rational mappings in one variable defined over the \(p\)-adic field \(\mathbb{C}_p\). We construct a mixing invariant probability measure which describes the asymptotic distribution of iterated preimages of a given point. This measure is supported on the Berkovich space \(P^1(\mathbb{C}_p)\) associated to \(\mathbb{P}^1 (\mathbb{C}_p)\). We show that its support is precisely the Julia set of \(R\) as defined by Rivera-Letelier. Our results are based on the construction of a Laplace operator on real trees with arbitrary number of branching as done in [C. Favre and M. Jonsson, The valuative tree, Lecture Notes in Mathmatics. Berlin etc.: Springer-Verlag, in press].

MSC:

37F10	Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
32H50	Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables
11S85	Other nonanalytic theory

Keywords:

iteration; \(p\)-adic field; equidistribution theorem; rational mappings; mixing invariant probability measure; asymptotic distribution; Julia set

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